<!DOCTYPE html>
<html>
<head>
<script>
var mQ = new Array();
var mQShow = 15; // message queue elements to show, negative no messages, 0 all
var msgBox = false;

function dmsg( message ){
	//if ( mQShow < 0 ) return;
	//if ( msgBox )
		msgBox = document.getElementById("dmsg"); // has the debug message been created?
	if ( msgBox ) { // element exists to add message to queue and show last mQShow message
		// add message to queue
		mQ[mQ.length] = message + "<br />";
		// build message text from message queue, display mQShow messages
		var mText = "", mCount = mQ.length - mQShow;
		if (mCount <= 0 ) mCount = 0;
		// append message queue to full box text
		for ( var i = mCount, len = mQ.length; i < len; i++ ){
			mText += mQ[i];
		}
		// set the message box
		msgBox.innerHTML = mText;
	} else { // queue message, italics because it cannot be displayed yet
		mQ[mQ.length] = "<i>" + message + "</i><br />";
	}
}
dmsg("debugging message on");
</script>
<!-- Standard content -->
<script src="./js/jQuery/jquery-1.8.js" type="text/javascript"></script> <!-- jQuery v1.8 local source -->
<script src="./js/standardComponents.js" type="text/javascript"></script> <!-- import standard teaching elements -->
	<!-- careful using id= make sure you do not use jQuery selectors, '.','()','[]','*','#' -->
<script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML" type="text/javascript"></script>
<script src="./js/standardEnumeration.js" type="text/javascript"></script>
<script src="./js/webColor.js" type="text/javascript"></script>
<script src="./js/geogebraWrapper.js" type="text/javascript"></script>
<script src="./js/quizTools.js" type="text/javascript"></script>
<script src="./js/en/keywords.js" type="text/javascript"></script>

<!-- Style selectors -->	
<!--<link rel="stylesheet" href="./css/standard.css" type="text/css"></style>-->
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
	extensions: ["tex2jax.js","TeX/AMSmath.js","TeX/AMSsymbols.js"],
	jax: ["input/TeX", "output/HTML-CSS"],
	tex2jax: {
		 inlineMath: [ ["$","$"], ["\\(","\\)"] ],
		 displayMath: [ ["$$","$$"], ["\\[","\\]"] ],
		 processEscapes: true
	},
	"HTML-CSS": { availableFonts: ["TeX"] }
});
</script>
<script>
$(document).ready(function(){
	indexPage();
	init();
	commentParse["ggb"] = GeogebraParse;
	//commentParse["quiz"] = QuizParser;
});
</script>
</head>
<body>
<!-- ggb init(file=./plugin/IdentSinAdd.ggb;img=./image/IdentSinAdd\1.png) -->
<div id="toc" class="toc">&nbsp;</div>
<div id="dmsg" class="dmsg">&nbsp;</div>
<h1>Proof of Identities: $\sin$ Addition Identities</h1>
<ul>
	<li class="objective prior">to be able to define and use trigonometric ratios, $\sin$ and $\cos$</li>
	<li class="objective">to understand how to calculate the $\sin$ of the sum of two angles</li>
	<li class="objective">to recognise that the equation is an identity</li>
	<li class="objective next">to use the identities to resolve sums of $\sin$ and $\cos$ <em>with same period</em> into a single $\sin$ or $\cos$ function</li>
</ul>
<h2>$\sin( \alpha + \beta )$</h2>
<table>
<tbody>
<tr>
<td valign="center">
<!-- 
The above is intended to create a Geogebra applet with the named file, the other options
can be set by using the "pass through" ggb mode. ggb <command>
The img tag is the format used to find pictures, if java is not available, a regex placeholder
for the img tag subsequent to geogebra ids
WL: This may get changed to allow svg to be used instead of static pictures
-->
<object type="application/x-java-applet" classid="java:geogebra.GeoGebraApplet" height="400" width="550" id="ggb" archive="./plugin/Geogebra4/geogebra.jar">
	<param name="filename" value="./mathggb/IdentSinAdd.ggb" />
	<param name="cache_archive" value="geogebra.jar, geogebra_main.jar, geogebra_gui.jar, geogebra_cas.jar, geogebra_algos.jar, geogebra_export.jar, geogebra_javascript.jar, jlatexmath.jar, jlm_greek.jar, jlm_cyrillic.jar, geogebra_properties.jar" />
	<param name="cache_version" value="4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0" />
	<param name="framePossible" value="false" />
	<param name="showResetIcon" value="false" />
	<param name="showAnimationButton" value="true" />
	<param name="enableRightClick" value="false" />
	<param name="errorDialogsActive" value="true" />
	<param name="enableLabelDrags" value="false" />
	<param name="showMenuBar" value="false" />
	<param name="showToolBar" value="false" />
	<param name="showToolBarHelp" value="false" />
	<param name="showAlgebraInput" value="false" />
	<param name="useBrowserForJS" value="true" />
	<param name="allowRescaling" value="true" />
</object>
<!--  
<applet name="ggbApplet" code="geogebra.GeoGebraApplet" archive="geogebra.jar" codebase="./plugin/Geogebra4/" id="ggb"
	align="top" width="600" height="480">
	<param name="filename" value="IdentSinAdd.ggb" />
	<param name="java_arguments" value="-Xmx512m -Djnlp.packEnabled=true" />
	<param name="cache_archive" value="geogebra.jar, geogebra_main.jar, geogebra_gui.jar, geogebra_cas.jar, geogebra_algos.jar, geogebra_export.jar, geogebra_javascript.jar, jlatexmath.jar, jlm_greek.jar, jlm_cyrillic.jar, geogebra_properties.jar" />
	<param name="cache_version" value="4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0, 4.0.4.0" />
	<param name="framePossible" value="false" />
	<param name="showResetIcon" value="false" />
	<param name="showAnimationButton" value="true" />
	<param name="enableRightClick" value="false" />
	<param name="errorDialogsActive" value="true" />
	<param name="enableLabelDrags" value="false" />
	<param name="showMenuBar" value="false" />
	<param name="showToolBar" value="false" />
	<param name="showToolBarHelp" value="false" />
	<param name="showAlgebraInput" value="false" />
	<param name="useBrowserForJS" value="true" />
	<param name="allowRescaling" value="true" />
</applet>
-->
</td>
<td valign="top">
<p>What is the $\sin$ ratio for $ \angle \alpha $?
<span class="show">
	<!-- reveal (OBC) -->
	<!-- ggb highlight(OB,BC) --> [show]</span></p>
<p class="reveal OBC">$$\sin{\alpha} = {{BC} \over {OB}} \implies {BC} = {OB} \sin{\alpha}$$</p>
<p class="reveal OBC">What about for $ \angle \beta $?
<span class="show">
	<!-- reveal (OAB) -->
	<!-- ggb highlight(OA,AB) --> [show]</span></p>
<p class="reveal OAB">$$\sin{\beta} = {{AB} \over {OA}} \implies {AB} = {OA} \sin{\beta}$$</p>
<p class="reveal OAB">We also will need the $\cos$ function
$$\cos{\beta} = {{OB} \over {OA}} \implies {OB} = {OA} \cos \beta$$</p>
<p class="reveal OAB">We now need to construct a single opposite side for the sum of the angles $\alpha$ and $\beta$
<span class="show">
	<!-- reveal (AXBY) -->
	<!-- ggb highlight()=off;show(AX,BY,X,Y,r_2);hide(r_0,r_1); --> [show]</span></p>
<p class="reveal AXBY">So we now get</p>
<p class="reveal AXBY">$$\sin (\alpha + \beta) = {{AX} \over {OA}} = {{AY + XY} \over {OA}} = {{AY} \over {OA}}+{{YX} \over {OA}}$$
$$ \implies {OA} \sin (\alpha + \beta) = {AY} + {XY} $$</p>
<p class="reveal AXBY">Now let us analyse angles; $OC$ is parallel to $YB$ with a transverse line, $OB$.<span class="show">
	<!-- reveal (ParAng) -->
	<!-- ggb highlight(OC,BY,OB);hide(r_2) -->[show]</span></p>
<p class="reveal ParAng">$ \angle {BOC} $ is alternate, to $ \angle {OBY} $ hence are <em>equal</em>.<span class="show">
	<!-- reveal (Sim_1) -->
	<!-- ggb show(a_1);highlight(r_0) -->[show]</span></p>
<p class="reveal Sim_1">The angle $ \angle ABY $ is $90^{\circ} - \alpha $, hence we can deduce $ \angle YAB $ is $\alpha$.
<span class="show">
	<!-- reveal (FTrig_1) -->
	<!-- ggb show(a_2);hide(a_1);highlight()=off -->[show]</span></p>
<p class="reveal FTrig_1">So we can find $AY$ in terms of $\alpha$ 
<span class="show">
	<!-- reveal (FTrig_2) -->
	<!-- ggb highlight(AY,AB) -->[show]</span></p>
<p class="reveal FTrig_2">$$\cos \alpha = {{AY} \over {AB}} \implies {AY} = {AB} \cos \alpha $$
<span class="show">
	<!-- reveal (FTrig_3) -->
	<!-- ggb highlight(XY,BC) -->[show]</span></p>
<p class="reveal FTrig_3">By parallel edges of a rectangle $XYBC$:$${XY} = {BC}$$
<span class="show">
	<!-- reveal (Conc_1) -->
	<!-- ggb highlight()=off -->[show]</span></p>
<p class="reveal Conc_1">Bringing these elements together:
$${OA} \sin (\alpha + \beta) = {AY} + {XY}$$
$${AY} = {AB} \cos \alpha ; {BC} = {XY} $$
$$\implies {OA} \sin (\alpha + \beta) = {AB} \cos \alpha + {BC} $$
Further substitutions from above:
<span class="show"><!-- reveal (Conc_2) -->[show]</span></p>
<p class="reveal Conc_2">$${AB} = {OA} \sin \beta ; {BC} = {OB} \sin \alpha; {OB} = {OA} \cos \beta$$
$${OA} \sin (\alpha + \beta) = {OA} \sin \beta \cos \alpha + {OA} \cos \beta \sin \alpha$$
</td>
</tr>
</tbody>
</table>

<h3>Positive Identity</h3>
<span class="show"><!-- reveal (BaseIdent,Neg_1) -->[show]</span>
<p class="reveal" id="BaseIdent">$$ \sin( \alpha + \beta ) \equiv \sin \alpha \cos \beta + \cos \alpha \sin \beta$$</p>

<h3>Negative Identity</h3>
<p class="reveal" id="Neg_1">Now it we consider when $\beta = -\delta$ and $\delta$ is <em>negative</em>:
<span class="show"><!-- reveal (Neg_2) -->[show]</span></p>
<p class="reveal" id="Neg_2">$$\sin( \alpha + \delta ) = \sin( \alpha - \beta )$$
$$ \implies \sin( \alpha - \beta ) = \sin \alpha \cos -\beta + \cos \alpha \sin -\beta$$
<span class="show"><!-- reveal (Neg_3) -->[show]</span></p>
<p class="reveal Neg_3">Because:
<ul class="reveal Neg_3">
	<li>$\cos$ is an <em>even function</em>, $\cos(-\beta)=\cos(\beta)$, and</li>
	<li>$\sin$ is an <em>odd function</em>, $\sin(-\beta)=-\sin(\beta)$</li>
</ul></p>
<p class="reveal Neg_3">
$$ \implies \sin( \alpha - \beta ) = \sin \alpha \cos \beta + \cos \alpha (-)\sin \beta$$
$$ \implies \sin( \alpha - \beta ) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$$
</p>

<h2>Sum/Difference Identity</h2><span class="show"><!-- reveal (CorePosNeg) -->[show]</span>
<p class="reveal CorePosNeg important">
$$ \sin( \alpha \pm \beta ) \equiv \sin \alpha \cos \beta \pm \cos \alpha \sin \beta$$
</p>

<h2>Double Angle Identity</h2>
<p class="reveal CorePosNeg">
What happens if we make $\alpha$ = $\beta$?<span class="show"><!-- reveal (2AngPosNeg) -->[show]</span></p>
<p class="reveal 2AngPosNeg">
$$\sin ( \alpha \pm \alpha ) = \sin \alpha \cos \alpha \pm \cos \alpha \sin \alpha$$
</p>
<p class="reveal 2AngPosNeg">
So lets consider the negative version first<span class="show"><!-- reveal (2AngNeg) -->[show]</span></p>
<p class="reveal 2AngNeg">
$$\sin ( \alpha - \alpha ) = \sin (0) = 0 = \sin \alpha \cos \alpha - \cos \alpha \sin \alpha = 0$$</p>
<p class="reveal 2AngNeg">
Not very useful, is it! So what about the positive version?<span class="show"><!-- reveal (2AngPos) -->[show]</span></p>
<p class="reveal 2AngPos">
$$\sin ( \alpha + \alpha ) = \sin \alpha \cos \alpha + \cos \alpha \sin \alpha$$</p>
<p class="reveal 2AngPos important">$$\sin (2 \alpha) \equiv 2 \sin \alpha \cos \alpha$$</p>

<h2>Sum to Product Identities</h2>
<p class="reveal CorePosNeg">
If we take both forms of the Sum/Difference Identity we get ($\alpha \to A$ and $\beta \to B$):
<span class="show"><!-- reveal (S2P1,D2P1) -->[show]</span></p>
<table width="100%">
<tbody>
<tr>
<td width="50%">
<h3 class="reveal S2P1">Summing</h3>
<p class="reveal S2P1">
$$\sin ( A + B ) = \sin A \cos B + \cos A \sin B$$
$$\sin ( A - B ) = \sin A \cos B - \cos A \sin B$$
</p>
<p class="reveal S2P1">
If we sum the functions we get:
<span class="show"><!-- reveal (S2P2) -->[show]</span></p>
<p class="reveal S2P2">
$$\sin ( A + B ) + \sin ( A - B ) = 2 \sin A \cos B$$
</p>
<p class="reveal S2P2">
Redefining the variables on the left, 
$$\alpha = A + B ; \beta = A - B$$
<span class="show"><!-- reveal (S2P3) -->[show]</span></p>
<p class="reveal S2P3">
$$\alpha + \beta = ( A + B ) + ( A - B ) = 2 A$$
$$\alpha - \beta = ( A + B ) - ( A - B ) = 2 B$$
<span class="show"><!-- reveal (S2P4) -->[show]</span></p>
<p class="reveal S2P4 important">
$$\sin \alpha + \sin \beta = 2 \sin {{\alpha + \beta} \over 2} \cos {{\alpha - \beta} \over 2}$$
</td>
<td width="50%">
<h3 class="reveal D2P1">Differences</h3>
<p class="reveal D2P1">
$$\sin ( A + B ) = \sin A \cos B + \cos A \sin B$$
$$\sin ( A - B ) = \sin A \cos B - \cos A \sin B$$
</p>
<p class="reveal D2P1">
If we subtract the functions we get:
<span class="show"><!-- reveal (D2P2) -->[show]</span></p>
<p class="reveal D2P2">
$$\sin ( A + B ) - \sin ( A - B ) = 2 \cos A \sin B$$
</p>
<p class="reveal D2P2">
Redefining the variables on the left, 
$$\alpha = A + B ; \beta = A - B$$
<span class="show"><!-- reveal (D2P3) -->[show]</span></p>
<p class="reveal D2P3">
$$\alpha + \beta = ( A + B ) + ( A - B ) = 2 A$$
$$\alpha - \beta = ( A + B ) - ( A - B ) = 2 B$$
<span class="show"><!-- reveal (D2P4) -->[show]</span></p>
<p class="reveal D2P4 important">
$$\sin \alpha - \sin \beta = 2 \cos {{\alpha + \beta} \over 2} \sin {{\alpha - \beta} \over 2}$$
</p>
</td>
</tr>
</tbody>
</table>

<hr />
<dl>
<dt>even function</dt><dd>A function where the negative domain has the same value as the positive domain; $f(-x)=f(x)$</dd>
<dt>odd function</dt><dd>A function where the negative domain has the opposite value to the positive domain; $f(-x)=-f(x)$</dd>
</dl>
<script type="text/javascript">dmsg("flush");</script>
<noscript>
<p><center><em>The webpage you are trying to view requires javascript and java plugins. Look for flat notes on this topic, but these will require Adobe Acrobat. Java sections are intended to be given a static page option in time (for iPad and Android users).</em></center></p>
<p><center>If you have turned off javascript to improve browser security or performance, please turn it on so you can view the webpage as was intended.</center></p>
</noscript>
</body>
</html>